Page 154 - Intro Predictive Maintenance

Vibration Monitoring and Analysis 145

M 2 ˙˙
(
X 2 =- K X 2 + K X 1 - X 2 )
3
2
g c
or
M 2 ˙˙
)
X 2 + ( K 2 + K X 2 - K X 1 = 0
3
3
g c
If we assume that the masses, M 1 and M 2, undergo harmonic motions with the same
frequency, w, and with different amplitudes, A 1 and A 2, their behavior can be repre-
sented as:
X 1 = A 1 sin w t
()
()
X 2 = A 2 sin w t
By substituting these into the differential equations, two equations for the amplitude

A 1
ratio, , can be found:
A 2
-
A 1 K 3
=
M 1
A 2 2
w - K 1 - K 3
g c
and

M 2 2
w - K 2 - K 3
A 1 g c
=
-
A 2 K 3
For a solution of the form we assumed to exist, these two equations must be equal:

M 2 2
w - K 2 - K 3
-K 3 g c
=
M 1 2 -K 3
w - K 1 - K 3
g c
or

ÏK 1 + K 3 K 2 + K 3 ¸ KK 2 + K K 3 + KK 3
1
2
1
2
4
w - w Ì + ˝ + MM = 0
2
Ó Mg c Mg c ˛ 1 2
1
2
g c
2
This equation, known as the frequency equation, has two solutions for w . When sub-
stituted in either of the preceding equations, each one of these gives a definite value